# inverse Gamma Distribution and LogNormal Distribution can both discribe the data, is that a coincidence?

I am observing the fluxes of source and I am trying to learn something from it's distribution of fluxes.

When I histogram my data, I can perfectly describe my data with a lognormal distribution. That was in fact what I had expected, but I was still curious if I could find other description for the flux distribution.

From a physical model point of few, a scenario where the fluxes are distributed according to the gamma distribution would make sense (for which I have a mind-picture of "events for which a waiting time is relevant").

To my disappointment the Gamma Distribution did a very poor job in describing the data. In a trail and error approach, I went forth and fitted the flux distribution with the Inverse Gamma Distribution. I found it to be a perfect fit for my data, even better than LogNormal distribution (not significantly in a chi^2 sense though).

I am now a bit dumb-struck:

First, because I don't have a "mind picture" for the inverse gamma distribution. Does it ever occur in nature other than being a vehicle for Bayesian interference? Could you provide such a "mind picture"?

Second, is there any reason for the Inverse Gamma distribution to fit some exponential data, other than it being a "flexible function" (flexible in a sense that I could probably fit a polynomial of some order to may data as well).

• Presumably a "flux" is a rate of something per unit time. Re-express them in terms of time per unit total flux: that is, take their reciprocals. That doesn't change the meaning of the data, but if originally they had an inverse-gamma distribution, the reciprocals have a gamma distribution; and if originally they had a lognormal distribution, then they still have a lognormal distribution. Thus, your situation no longer seems terribly exotic: you are wondering under what circumstances data might be approximated by either a gamma or lognormal distribution. – whuber Jun 10 at 21:03
• Uh, this sounds promising! The flux in the light curve is in units of energy at a given time, but I guess this makes the histogram in units of 1/flux. Could you provide me with some further reading on what reciprocals mean in statstics (that is a badly worded question, but I is it clear what I need? just some further reading, or a "real life example"? Thanks already. – Sebastiano1991 Jun 10 at 21:18
• Reciprocal flux has a natural interpretation as the amount of time needed to transmit a unit of energy. Good books on exploratory data analysis include a lot of discussion about "re-expressing" or "transforming" variables. John Tukey's book EDA is a notable resource. You can find a lot about this on our site, too, under the name Box-Cox transformation: the reciprocal is essentially the same as a Box-Cox transformation of power -1. – whuber Jun 10 at 21:46
• Does any simple distribution really "occur in nature"? In general, they're just approximations (sometimes really, really good ones). I've used inverse Gamma distributions a few times (e.g. on insurance claim sizes where the lognormal wasn't skewed enough; it worked remarkably well for some particular kinds of data) and I've seen even more strongly skewed inverse Gaussian distributions used a few times (e.g. for earthquakes) but in each case using it doesn't mean that it was anything more than a useful approximation. – Glen_b -Reinstate Monica Jun 11 at 1:12
• i.e. consider the advice of George Box ... "Remember that all models are wrong. The practical question is how wrong do they have to be to not be useful." $\:$ Now, to differentiate between a lognormal and something more right skew (including the inverse gamma), look at the log. If it's still right skew, you can rule out the two-parameter lognormal. Now for your inverse gamma (as long as the shape parameter isn't too small), take the cube root of the reciprocal; if that's not pretty close to normally distributed, the inverse gamma would be ruled out. – Glen_b -Reinstate Monica Jun 11 at 1:15